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mathematics
statistics

Quantitative Methods For Business 5th Edition Donald Waters - Solutions

Amalgamated Engineering makes two kinds of gearbox - manual and automatic. There are four stages in the production of these, with details of times needed and weekly availabilities given below. The company makes a profit of £64 on each manual gearbox sold and £100 on each automatic
Figure shows a printout from a linear programming package. Explain what these results show. How could the format of the results beimproved?
Linear programming is part of a family of methods for solving different types of constrained optimisation problems. Members of this family include integer, zero-one, non-linear and goal programming. Describe the features of other kinds of programming, and the kinds of problem that they tackle.
The demand in all branches of banks varies during the day. The AIBC International branch in Toronto has a peak demand for domestic transactions around lunchtime. When this is translated into the number of employees needed in the branch, it gives the following pattern. This demand is met by a
1. The Gamblers' Press finds schemes like these four almost every day, and often publishes articles on them. What do such schemes have in common? If you were asked to write an article about these four schemes, what would you say? You might start by explaining why the four schemes mentioned do not
An office has the following types of employees.If one person from the office is chosen at random, what is the probability that the person is: (a) A male administrator, (b) A female manager, (c) Male, (d) An operator, (e) Either a manager or an administrator, (f) Either a female administrator or a
A quality control test has five equally likely outcomes - A, B, C, D and E.(a) What is the probability of C occurring? (b) What is the probability of A or B or C occurring? (c) What is the probability that neither A nor B occur?
Four mutually exclusive events A, B, C and D have probabilities of 0.1, 0.2, 0.3 and 0.4 respectively. What are the probabilities of the following occurring? (a) A and B, (b) A or B, (c) Neither A nor B, (d) A and B and C, (e) A or B or C, (f) None of A, B or C?
If you choose a card at random from a complete pack, what is the probability that it is? (a) An ace(b) A heart(c) An ace and a heart(d) An ace or a heart(e) Neither an ace nor a heart?
Bert Klassen schedules three calls for a particular day, and each call has a probability of 0.5 of making a sale. What are his probabilities of making? (a) Three sales(b) Two or more sales, (c) No sales?
There are 20 people in a room. What is the probability that they all have different birthdays?
If P (a) = 0.4 and P(b/a) = 0.3, what is P(a and b)? If P(b) = 0.6, what is P(a/b)?
The probabilities of two events X and Y are 0.4 and 0.6 respectively. The conditional probabilities of three other events A, B and C occurring, given that X or Y has already occurred, are:What are the conditional probabilities of X and Y occurring, given that A, B or C has alreadyoccurred?
Kenny Lam works in a purchasing department that buys materials from three main suppliers, with X supplying 35% of the department's needs, Y supplying 25% and Z the rest. The quality of the materials is described as good, acceptable or poor with the following proportions from each supplier:What
Data collected from Cape Town shows that 60% of drivers are above 30 years old. 5% of all the drivers over 30 will be prosecuted for a driving offence during a year, compared with 10% of drivers aged 30 or younger. If a driver has been prosecuted, what is the probability they are 30 or younger?
The Gamblers' Press case study showed some misleading uses of statistics. Unfortunately this is fairly common. Find some other example where statistics are used to give the wrong impression - either intentionally or unintentionally.
Marius Gensumara found that during a typical period the following numbers of people did not turn up to work at his company in consecutive days:13, 16, 24, 21, 15, 23, 15, 26, 25, 11,10, 24, 27, 30, 15, 31, 25, 19, 15, 27He wanted to improve this and introduced a new scheme of
1. Operations really need a study to see why there is variability, how much is acceptable, what its effects are, how it can be reduced, what benefits this will bring and so on. Such a study needs funding - and your job is to write an initial proposal for this funding, including a detailed proposal
Find the probability distribution of this set ofobservations:
Find the values of nCr and nPr when (a) r = 5 and n = 15, (b) r = 2 and n = 10, (c) r = 8 and n = 10.
An open-plan office has ten desks. If ten people work in the area, how many different seating arrangements are there? If two people leave, how many arrangements are there?
A salesman has 12 customers to visit each day. In how many different ways can he visit the customers? One day he has time to visit only eight customers. In how many different ways can he select the eight? As the salesman has to travel between customers, the order in which his visits are scheduled
A binomial process has a probability of success of 0.15. If eight trials are run, what are the mean number of successes and the standard deviation? What is the probability distribution for the number of successes?
In a town, 60% of families are known to drive European cars. In a sample of 10 families, what is the probability that at least 8 drive European cars? In a sample of 1,000 families, what is the probability that at least 800 drive European cars?
Norfisk Oil is drilling some exploratory wells on the mainland of Norway. The results are described as either a 'dry well' or a 'producer well'. Past experience suggests that 10% of exploratory wells are producer wells.If the company drills 12 wells, what is the probability that all 12 are producer
100 trials are run for a Poisson process. If the probability of a success is 0.02, what are the mean number of successes and the standard deviation? What is the probability distribution for the number of successes? What is the probability of at least six successes?
During a typical hour an office receives 13 phone calls. What is the distribution of phone calls in a 5-minute period?
During a busy period at an airport, planes arrive at an average rate of 10 an hour. What is the probability distribution for the number of planes arriving in an hour?
A machine makes a product, with 5% of units having faults. In a sample of 20 units, what is the probability that at least 1 is defective? In a sample of 200 units, what is the probability that at least 10 are defective?
A set of observations follow a Normal distribution with mean 40 and standard deviation 4. What proportions of observations have values?(a) Greater than 46,(b) Less than 34,(c) Between 34 and 46,(d) Between 30 and 44,(e) Between 43 and 47?
A large number of observations have a mean of 120 and variance of 100. What proportion of observations is?(a) Below 100(b) Above 130(c) Between 100 and 130(d) Between 130 and 140(e) Between 115 and 135?
The number of meals served in a week at Cath's Café is Normally distributed with a mean of 6,000 and a standard deviation of 600. What is the probability that in a given week the number of meals served is less than 5,000? What is the probability that more than 7,500 meals are served? What is the
A service consists of two parts. The first part takes an average of 10 minutes with a standard deviation of 2 minutes; the second part takes an average of 5 minutes with a standard deviation of 1 minute. Describe how long it takes to complete the service. What is the probability that a customer can
Why are scheduling problems so difficult? Plane, bus and train timetables show expected schedules - so how do you think these are designed? Choose a convenient service and collect data to show how actual arrival times compare with expected times. What can you say about these results?
The number of people visiting a shop each working hour for the past week has been recorded as follows:What do these results show? Are they typical of the distribution of customer numbers at other shops? How would you set about collecting and analysing data to get more information about the
1. What information can James find from these figures? How can he use this information?2. How should he set about a complete survey of the fruit crop?In the 1920s Edward Filbert became the tenant of Kings Farm in Cambridgeshire. In 1978 his grandson James Filbert became the latest manager. In the
A production line makes units with a mean weight of 80 g and standard deviation of 5 g. What is the probability that a sample of 100 units has a mean weight of less than 79 g?
A machine makes parts with a variance of 14.5 cm in length. A random sample of 50 parts has a mean length of 106.5 cm. What are the 95% and 99% confidence intervals for the length of parts?
A frozen-food packer specifies the mean weight of a product as 200 g. The output is Normally distributed with a standard deviation of 15 g. A random sample of 20 has a mean of 195 g. Does this suggest that the mean weight is too low?
Hamil Sopa took a random sample of 60 invoices from his year's records. The mean value of invoices in this sample was £125.50 and the standard deviation was £10.20. What are the 90% and 95% confidence intervals for the mean value of all invoices?
Sheila Brown times 60 people doing a job. The mean time is 6.4 minutes, with a standard deviation of 0.5 minutes. How long would it take the population to do this job?
Wade (Retail) looked at a random sample of 100 invoices from a large population. Eight of these contained an error. What are the 90% and 95% confidence intervals for the proportion of invoices with errors?
A company wants to find the average weight of its products. A large initial sample shows the standard deviation of the weight is 20 g. What sample size would give a 95% confidence interval for the population that is (a) 10 g wide, (b) 8 g wide, (c) 5 g wide?
Last year a trial survey found that 65% of houses in Morrisey township had a computer. A follow-up survey wants to find the actual number of houses with a computer to within 3% with a 95% confidence interval. How many houses should it survey?
Henry Lom feels that the quantity of chocolates in a particular type of packet has decreased. To test this feeling he takes a sample of 40 packets and finds that the mean weight is 228 g with a standard deviation of 11 g. What is the weight Henry can be 95% confident the mean falls below? What are
BC's quality assurance programme chooses a random sample of 50 units and finds that 12 are defective. What is the number of defectives they can be 95% confident that the population mean is below? What is the number of defectives they can be 95% confident that the population mean is above? How do
A survey of 20 items in a sales ledger has a mean value of €100 and standard deviation of €20. What is the 95% confidence interval for the population of items? What is the 99% confidence interval?
The time taken for a sample of eight pieces of equipment to do a task has a mean of 52 minutes and a standard deviation of 18 minutes. What is the 90% confidence interval for the mean time of all the equipment to do the task? What is the 95% confidence interval? If the same results had been found
For small samples we have to use the t-distribution instead of the Normal distribution. How would you describe - and measure - the differences between these two distributions?
The central limit theorem gives a fundamental result for sampling - that large samples, or any samples from a Normally distributed population, are Normally distributed. Test this result to see that it really works. For this you can:Use a spreadsheet to generate a population of random numbersDraw a
1. What could the milk inspectors report about their findings? What follow-up action could they recommend? Are there any improvements they could make to their data collection and analysis?Willingham Consumer Protection Department (WCPD) is responsible for administering all weights and measures laws
The mean wage of people living in Alto Canillas is said to be £400 a week with a standard deviation of £100. A random sample of 36 people was examined. What is the acceptance range for a 5% significance level? What is the acceptance range for a 1% significance level?
The weight of packets of biscuits is claimed to be 500 g. A random sample of 50 packets has a mean weight of 495 g and a standard deviation of 10 g. Use a significance level of 5% to see whether or not the data from the sample supports the original claim.
Hamil Coaches Ltd say that their long-distance coaches take 5 hours for a particular journey. Last week a consumer group tested these figures by timing a sample of 30 journeys. These had a mean time of 5 hours 10 minutes with a standard deviation of 20 minutes.What report can the consumer group
A food processor specifies the mean weight of a product as 200 g. A random sample of 20 has a mean of 195 g and a standard deviation of 15 g. Does this evidence suggest that the mean weight is too low?
An emergency breakdown service suggests that 50% of all drivers are registered with their service. A random sample of 100 people had 45 who were registered. Does this sample support the original claim?
Quality Managers at CentralGen say that 12% of the letters they post contain errors. A sample of 200 letters was checked and 31 of them contained errors. What do these results suggest?
Health service managers say that doctors should not spend more than 2 hours a day doing paperwork. A sample of 40 doctors spends an average of 2 hours 25 minutes a day doing paperwork, with a standard deviation of 55 minutes. What does this show?
A mobile phone has an advertised life of 30,000 hours. A sample of 50 phones had a life of 28,500 hours with a standard deviation of 1,000 hours. What can you say about the advertisements?
Dorphmund Industries have two similar factories. There is some disagreement, because people working in each factory think those in the other factory are getting higher wages.A sample of wages was taken from each factory with the following results:Sample 1: size = 45, mean = $250, standard deviation
A car manufacturer says that its cars cost €500 a year less to maintain than those of its competitors. To test this, a consumer group found the cost of maintaining 10 cars for a year, and the mean saving was €79 with a standard deviation of €20. What does this say about the manufacturer's
Five factories reported the following numbers of minor accidents in a year:Does this suggest that some factories have more accidents thanothers?
The following figures show the number of defective components supplied each day by a factory. Does this data follow a binomialdistribution?
The number of road accident victims reporting to a hospital emergency ward is shown in the following table. Do these figures follow a Poissondistribution?
Do the following figures follow a Normaldistribution?
Figure shows a spreadsheet doing the calculations for a t-test on the mean of two samples. Explain the results and check the calculations. How could you improve theformat?
This chapter mentioned several examples of automatic systems that implicitly include hypothesis tests - including airport security systems e-mail spam filters Internet search results, automatic recorders of earthquakes, burglar alarms and roadside speed cameras.What other examples can you find? How
Hypothesis testing comes in many different forms, and it always seems to involve judgement. This makes it difficult to design a package that automatically takes data and does an appropriate hypothesis test. Do a small survey to see what facilities statistical packages have for hypothesis testing.
Supermarkets and other retailers often claim that they offer the lowest prices in their area. How can you check their claims? Collect some data from competing stores and analyse the results. What conclusions can you reach?
Often a hypothesis may seem 'obvious', but on closer examination there is no evidence to support it. Find some real examples of this effect. What are the consequences?
1. What would a decision tree of the farmers' problem look like?2. What are their best choices and expected costs?Newisham has a population of about 30,000. It traditionally got its water supply from the nearby River Feltham but increasing quantities of water were being extracted from the river by
O'Brian's pub on the seafront at Blackpool notices that its profits are falling. The landlord has a number of alternatives for increasing his profits (attracting more customers, increasing prices, getting customers to spend more etc.) but each of these leads to a string of other effects. Draw a map
Choose the best alternative in the following matrix of gains.Event GainA .......... 100B .......... 950C .......... −250D .......... 0E .......... 950F .......... 500
Use the Laplace, Wald and Savage decision criteria to select alternatives in the following matrices. What results would you get for other decision criteria?(a) Cost matrix(b) Gainsmatrix
Figure shows a printout from a program which does the calculations for decision criteria.Describe the criteria that it uses and design your own spreadsheet to check the results.What results would other criteria give?
Which is the best alternative in the following gains matrix? Would this decision change using a utility function U(x) =ˆšx?
GKR WebSpace can launch one of three versions of a new product, X, Y or Z. The profit depends on market reaction and there is a 30% chance that this will be good, a 40% chance it will be medium and a 30% chance it will be poor. Which version should the company launch with the profits given in the
Schwartz Transport owns a lorry with a one-year old engine. It has to decide whether or not to replace the engine at a cost of ‚¬2,000. If it does not replace the engine, there is an increased chance that it will break down during the year and the cost of an emergency replacement is
Wilshaw Associates is considering launching an entirely new service. If the market reaction to this service is good (which has a probability of 0.2) they will make $30,000 a month; if market reaction is medium (with probability 0.5) they will make $10,000; and if reaction is poor (with probability
A television company has an option on a new six-part series. They could sell the rights to this series to the network for £100,000 or they could make the series themselves. If they make the series themselves, advertising profit from each episode is not known exactly but could be £15,000
We have described several formats for presenting decisions - problem maps, payoff matrices and decision trees. But these are not the only options. What other formats are available? Find some examples where different formats have been used in practice.
Some software packages - or special add-ins for spreadsheets - draw decision trees automatically. Do a small survey to see what features these contain.You can also draw a decision tree on a standard spreadsheet, as illustrated in Figure. This uses the DRAW options for drawing the skeleton of the
1. Do you think the General Manager's view is reasonable? What about Jurgen Hansmann's reaction?2. How effective is quality control at Bremen Engineering?3. Do you think the product quality needs to be improved? How would you set about this?Jurgen Hansmann is the Quality Control Manager of Bremen
Amwal Corporation had the following costs (in thousands of pounds) over the past six years. Describe what has beenhappening.
Hung Gho Chan make a part on an assembly line. At one point they find that 2% of units are defective. It costs $1 to inspect each unit at this point, but the inspection would find only 70% of faults. If the faults are left, all parts will be found and corrected further down the line at a cost of
Sentinal Phoneback answers customer enquiries with telephone calls. When they timed a sample of 40 calls, they found a mean duration of 14.9 minutes and a standard deviation in duration of 2 minutes. What are the 95% and 99% confidence intervals for the true length of calls?
Eriksonn Catering says that its suppliers should send at most 2% of units that do not meet its 'outstanding' standard of quality. It receives a large shipment and takes a sample of 100 units. The company wants to be 95% sure that a rejected batch is really unsatisfactory. What criteria should it
Carn Bay Components make batches of a basic product in Toronto and transfer it to their main manufacturing plant in Chicago. When the product is made, an acceptance quality level of 1% defective is used, but transferred batches are allowed a maximum of 4% defective. The company accept a 5% risk of
A service provider checks 24 samples of 200 clients to see whether or not they are giving an acceptable level of service. The numbers of unsatisfactory results were as follows.Draw control charts with 95% and 99% confidence limits on theprocess.
Gunta Hans took 30 samples of 15 units from a process. The average sample range for the 30 samples is 1.025 kg and the average mean is 19.872 kg. Draw control charts for the process.
Pioneer Remedial found that a particular repair takes a mean time of 75.42 minutes with a standard deviation of 2.01 minutes. If samples of eight are taken, find the control limits that include 99% of sample means if the process is working normally.
Find a product that you have been particularly pleased with. Describe the aspects of its quality that you like. How many of these aspects can you measure? How many people - from initial designers through to the person who delivered it - were involved in supplying this high-quality product? How
Quality management has undergone a revolution in recent years, with customers no longer willing to accept defective products. A key element of this has been the changing role of quality control. Describe, giving suitable examples, how this role has changed in practice.
A lot of software is available for quality control. The figure shows a printout from a simple package that takes data, suggests a sampling plan and shows the operating curve for this plan and the average outgoing quality. How does this compare with other software? What features do you think there
Despite the attention paid to quality management, many products still do not meet acceptable standards. Give some examples of products that you think are unsatisfactory. Why is this? What can be done to improve these products?
1. If you were the management consultant, how would you demonstrate the benefit of a new stock control system to Mr Templar?2. What information would you need for your demonstration?James Templar founded his own manufacturing company when he was 21 years old. He has continued to run it for the past
The demand for an item is constant at 100 units a year. Unit cost is £50, the cost of processing an order is £20 and holding cost is £10 a unit a year. What are the economic order quantity, cycle length and costs?
Beograd Inc. works 50 weeks a year and has demand for a part that is constant at 100 units a week. The cost of each unit is $200 and the company aims for a return of 20% on capital invested. Annual warehouse costs are 5% of the value of goods stored. The purchasing department costs $450,000 a year
Demand for an item is steady at 20 units a week and the economic order quantity has been calculated at 50 units. What is the reorder level when the lead time is (a) 1 week, (b) 2 weeks?
How would the results for Problem 18.1 change if the part could be supplied only at a fixed rate of 10 units a week? Would there be any benefit in reducing production to 5 units a week?

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